Haus, Hermann A.

Penfield, Paul L., Jr.

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IX. ELECTRODYNAMICS OF MOVING MEDIAProf. L. J. ChuProf. H. A. HausProf. P. Penfield, Jr.A. ENERGY CONSERVATION THEOREMI FOR THE PHONON MASER1. Linearized EquationsThe interaction of sound waves with optical-frequency waves in the presence of astrong optical maser beam has been analyzed by Kroll. 1 Bobroff2 uses the followinglinearized fluid equations:SaaHV x E = -C0- (1)V X H= E + (psE) (2)at K at saP = -Vp + VE (3)P t s s 2ap1 s -V 7 (4)K at sThe subscript 0 denotes time-average quantities; the velocity vs, mass density ps,and pressure ps are small perturbations of the corresponding quantities away from equi-librium; the constant y is the electrostrictive constant; and K is the bulk modulus ofthe fluid. While these equations are adequate for describing the interaction of the elec-tromagnetic and sound waves, they are approximate in two respects: (i) they have beenlinearized with respect to the velocity v and density p of the material, and (ii) certainterms that are small (in the ratio of sound speed to the speed of light, c) have been dis-regarded.2. General Nonrelativistic EquationsIt is helpful to consider a more complete form of these equations when deriving con-servation theorems because then the terms appearing in the theorem can be interpretedphysically more easily. In this report we shall consider equations that are exact exceptfor their lack of relativistic terms involving (i) the square of the material velocity2 2 3divided by c , and (ii) the ratio of electric energy density to pc . These equations areMaxwell's equationsX aH (5)Sx E = -_1A0 at-QPR No. 74(IX. ELECTRODYNAMICS OF MOVING MEDIA)8E aPVX H = E + + VX (PX ), (6)0 at atwhere P is the polarization density in the fluid; the force equation-t Pv + V pV= 7 =f-Vr + P VE + X (P V)h0H + +V( vP) X 0H, (7)where f is the force density in the medium, and the equation of conservation of mass- p + 7V pv = 0. (8)Here no linearization assumptions have been made. The term Tr is the fluid pressurecaused by a combination of hydrodynamic and electrostrictive effects,i = r(P, p)= p -w + (E+vX 0 H) * P, (9)where w, a function of p and P, is the energy density in the material because of hydro-dynamic and polarization effects. The change in w that is due to a change in density andpolarization density is given by/8w \  - _6w =(p 6p + (E+VXi 0 H) • 8P(_ w (E+v X oH) ) P. .= + w- - 6p + (E+vX H) • 6P. (10)By introducing the concept of an energy density, we have assumed that the medium isconservative, with a constitutive law of the formP = P(p, E+VX 40H). (11)A useful alternative form for the force equation (7) can be derived by use of somevector identities and (10) as follows:V +-+- = - OH [ (vP- H . (12)The other three equations then areapV7 pi -at (13)8H (14)VX E = -,,0 at (14)QPR No. 74(IX. ELECTRODYNAMICS OF MOVING MEDIA)aE aPV X (H-PX v) = E + (15)0 at atThis set of four equations (12-15) is ideally suited for derivation of power theoremsbecause on the left are spatial derivatives and on the right are time derivatives. Notethat these equations are exact, nonrelativistically, and have not been linearized.3. Equivalence of the Two Sets of EquationsTo prove that (1-4) are in fact approximate expressions obtained from (6-8), we con-sider the linearized expression for the constitutive law.P = 0 Xe (E+vX 0 H) + PE 0aeE. (16)The source term on the right-hand side of (6), to first order in perturbed density andvelocity, becomesaP + V × ( v) = (EOXeX)+ 8t EOoV X8t (XV) oxE + TPE +2 (CX fO) + V. E0 X (vE-Ev).(17)Whereas coupling of the acoustic system to the electromagnetic system requires reten-tion of terms of first order in density and velocity, the coupling of the electromagneticsystem to the acoustic system takes place in zero order in p and V. The force density(7) may then be written in the form= -Vps +-VE OPo + (EOX E XOH). (18)If we make the identificationaxP a0  = Y' (19)and disregard the last two terms in (17) and third term in (18), we obtain the sourceterm of (2) and force coupling term of (3). The terms that have been disregarded areusually of order cs/c (where c s is the speed of sound and c is the speed of light in themedium under consideration) as compared with the terms that have been retained.4. Conservation of EnergyThe law of conservation of energy can be derived very easily from (12-15). Dot-multiply (12) by pV, multiply (13) by the quantity that appears in parentheses on theQPR No. 74(IX. ELECTRODYNAMICS OF MOVING MEDIA)left-hand side of (12), dot-multiply (14) by (H-P Xv), and dot-multiply (15) by -E. Thenadd the four equations, to obtainr p - 1 2 -1 2 1 2S X Hv+v2 + v(w+r) - P(E 7) = -- pv2 +w E E + 0 H . (20)Z 2Here, we have evaluated aw/at by means of (10). This conservation equation is of a verysimple form. The terms under the divergence are the electromagnetic Poynting vector,the kinetic power-flow density, the flow of enthalpy (as augmented by the electrostrictivecontributions toward the pressure), and the rate of work per unit area by the stress ten-sor PE associated with the polarization of the material. On the right-hand side are thetime derivatives of several energy densities: the kinetic energy density, the energydensity w associated with the material, and the electric and magnetic energy densities.5. Other Conservation TheoremsA class of solutions of (12)-(15) exists for whichX - p = 0. (21)For these "irrotational" cases, a variety of other conservation theorems can be derivedfrom the basic equations, (12)-(15). Many such theorems for a similar physical system,the relativistic electron beam, have been derived and discussed previously,4 and all der-ivations are similar for the phonon-maser case, although the physical interpretation maybe different. Among the conservation principles that can be derived are small-signalpower theorems (both with arbitrary time dependence and sinusoidal time dependence),Manley-Rowe relations, small-signal energy theorems, expressions for power andenergy in waves, several large-signal power theorems, and variational principles.H. A. Haus, P. Penfield, Jr.References1. N. M. Kroll, Acoustic interaction of intense light beams and their bearing onmode control in high-power optical masers, Bull. Am. Phys. Soc. 9, 222 (March 23,1964).2. D. L. Bobroff, The Rate of Oscillation Build-up in the Phonon-Phonon ParametricBackward-wave Oscillator, Unpublished private memorandum, Raytheon Research Divi-sion, Waltham, Massachusetts.3. Cf. Quarterly Progress Report No. 70, Research Laboratory of Electronics,M. I. T., July 15, 1963, pp. 79-88.4. A. Bers and P. Penfield, Jr., Conservation principles for plasmas and relativ-istic electron beams, IRE Trans., Vol. ED-9, No. 1, pp. 11-26, January 1962.QPR No. 74

Electrodynamics of Moving Media

VII. ELECTRODYNAMICS OF MOVING MEDIAProf. L. J. ChuProf. H. A. HausProf. P. Penfield, Jr.A. FORCE DENSITY IN POLARIZABLE MATTER OBTAINED FROM THEPRINCIPLE OF VIRTUAL WORKThe force density in a moving, deforming polarizable material with a particularsingle-valued constitutive law relating the polarization P 0 and electric field E 0 in therest frame has been obtained previously from Hamilton's principle. The principle ofvirtual work offers an alternate way of obtaining the same result. The advantage of thisapproach is that it is less dependent on the assumed constitutive law. In fact, the resultsobtained are valid for any lossless dispersive polarizable fluid (e. g., gyroelectric fluid)in which the constitutive law relating P 0 and E 0 is an integro-differential equation intime.The principle of virtual work in a form that is suitable for the determination of forcedensities in continua starts from the law of energy conservation. We consider a movingand deforming medium and single out a moving and deforming volume V of the mediumenclosed by a surface S that is assumed to move along with the local velocity V(F, t) ofthe medium. Denote the tensor stress in the medium by t, the power flow passing thesurface by -, the energy density by w, and the power conversion density by c. Thelaw of energy conservation is thend . V+ g • da + w dv = f dv. (1)S S V VIt should be noted that g is the power passing through the surface S, such as electro-magnetic power or heat flow; the contribution to the time rate of change of the energyin V from the work of the surface stresses is contained in the first integral. As stated,Eq. 1 is of general validity and applies to dissipative or active systems and is not lim-ited to nonrelativistic motion as long as the various terms are all evaluated in the lab-oratory frame. The law of energy conservation is made into a principle of virtual workby restricting it to nondissipative systems, to small virtual velocities v, and to systemsfor which the quantities t, s, w, and 4 may be expressed in terms of the velocity v,and the stress to, energy, force, power conversion, and momentum densities w 0 , f,0' and G0 in the rest frame. Once such relations are known (or postulated) it is pos-sible to find tO f0, and G O in terms of s0' w 0, and 0. Some of these relations aresimple. Because t is a force per unit area, it is, in the nonrelativistic limit (to firstorder in V) equal to its rest-frame counterpartt = to (2)QPR No. 73(VII. ELECTRODYNAMICS OF MOVING MEDIA)The power flow density is also invariants = sO. (3)The power conversion density in the laboratory frame #, is related to 0 in the restframe byc = + f0 " v. (4)The second term takes into account the power conversion density contributed by the forcedensity f 0 working on a fluid moving with the velocity v.The relation between w and w0 is less intuitive if power flow and momentum densi-ties are present in the rest frame. An expression between w and w0 is obtained if onepostulates that the energy-momentum tensort+ vG 1 ricG -wtransforms as a four-tensor. One finds for the 4, 4-componentw = w 0 + 2 + G O ' (6)cBy introducing (2)-(6) in Eq. 1, using the particle (or mass) density n, one has, to firstorder in ',7 0 0 • (t- V) + f " V. (7)The derivatives are given with respect to the laboratory-frame coordinates. If one knowsthe expressions for g, w0 , and 50 in terms of the physical variables, if one makes useof the equations of motion of these physical variables, and if one retains only termsup to first order in V, the left-hand side of Eq. 7 becomes a first-order expression inv. Comparing the result with the right-hand side, one finds f%, GO, and t0 from a term-by-term indentification. This procedure is best illustrated by the following example.We apply the principle of virtual work to a dielectric fluid, without free charge orfree current. This fluid contains electric dipoles, with density (number per unit vol-ume) n and dipole moment P/n. The fluid energy (per unit volume) in the rest frameof the material may be assigned to each dipole: w0 /n 0 . The rate of change a/8t 0 (w0 /n 0 )is caused in two ways. Energy is fed into the dipole by the electric field in the restframe (where the subscript 0 on t denotes it as the rest-frame coordinate) when thedipole moment is changed. Thus, one contribution isQPR No. 73(VII. ELECTRODYNAMICS OF MOVING MEDIA)E a (8)This would be the only contribution if the microscopic and macroscopic electric fieldswere the same, and if there were no fluid pressure. Since the two are generally notthe same, there is an additional contribution that is proportional to the time rate ofchange of the density0 0 (9)n at 0The physical meaning of Tr 0 will emerge later on. The time rate of change of the energyper dipole is8 w0 a /P0 Tr0 anao  E a  (10)at0 n0 0 at0 n 0 2 at0The rate at which power (per unit volume) is fed into the fluid by the electromagneticfield can be obtained on a macroscopic basis from the Poynting theoremV • (EX H) + . E OE 2 0 H = -E • Jpol (11)where Jpol is the current density associated with the polarization,- 8aPJ + V X (PX V). (12)pol atTherefore, by applying this equation in the rest frame, the rest-frame power conversiondensity isS= E 0 JpolO. (13)The power-flow vector of the fluid system in the rest frame, sO, is zero. Since thepartial derivative with respect to the rest-frame time is equal to the substantive deriv-ative in the laboratory frame, the left-hand side of (7) is thus- d P-0 T 0 d n 0n (0E + dt E ' Jpolo = F. (14)0  dt n0 n0 dt 0 polOAll terms in (14) must be expressed in terms of the physical variables in the laboratoryframe because we want to make use of the equation of motion to bring (14) into the formof the right-hand side of (7), where all derivatives are given with respect to laboratory-frame coordinates. To eliminate the zero-order terms in Eq. 14, we use the transfor-mation laws, correct to first order in V,PO =  (15)0QPR No. 73(VII. ELECTRODYNAMICS OF MOVING MEDIA)E0 = E + v X j0 Hn0 = nJpolo= Jpol- pol +V x (PX-V) +- v * P +=+V (VP) - P V v.at at(18)In the rest frame, the law of conservation of particles is1 dn0V- v = - -n 0dtIf this is substituted in Eq. 18, we findd P VVpol = n0 dt n 0 0 " Vto first order in v. Now, use of Eqs. 14, 19, and 20 results inF= P 0  (VV) . EO - 70 v = v. (V 0 PoE 0 ) [(- 0 -POE) .v]and hence from Eq. 7 we recognize that the rest-frame stress tensor ist 0= r POE0,the rest-frame momentum density isG0 = 0,and the rest-frame force density isf0 = r - V . PE *When these quantities are transformed to an arbitrary reference frame, we findt = Tr06 - P(E+vX t0 H)S= Vi(O0 +w) - P(E V)w = w 0 +V. G,(21)(22)(23)(24)(25)(26)(27)where G is the relativistic momentum associated with the energy and stress in the fluid2 2G=v 2 V' Y 2 [T0 5- P(E +vX 0 )]. (28)c cNow let us add the stress-energy tensor for the polarization to the stress-energy tensorfor the fields. The result isQPR No. 73(16)(17)(19)(20)(VII. ELECTRODYNAMICS OF MOVING MEDIA)tot = ttot + VGtot 0 EE +- 0H) - (EE+P)E - P(vX 0o ) + VG (Z9)Gt = G + 2 (30)cstot = v0 + v(w 0 +V. G) - P(E v) + EX H (31)1 2 1 H2Wto= w + v - +- E +- H . (32)The divergence of the stress-energy tensor yields a force density of electromagneticoriginfe = (P -V)E + n d X 0H + V X (P -17) H - VT - n d ' (33)em dt n0 0 dt nwhich is in agreement with the force density for the same situation predicted by meansof Hamilton's principle. 2H. A. Haus, P. Penfield, Jr.References1. P. Penfield, Jr., Force of electromagnetic origin in fluids, Quarterly ProgressReport No. 70, Research Laboratory of Electronics, M. I. T., July 15, 1963, pp. 82-87.2. Ibid., page 86, Eq. 32.QPR No. 73

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Electrodynamics of Moving Media

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